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Integration by parts in 2D

  1. Jan 23, 2013 #1
    The integration by parts rule in two dimensions is
    [itex]\int_{Ω}\frac{\partial w}{\partial x_{i}} v dΩ = \int_{\Gamma} w v \vec{n} d\Gamma - \int_{Ω} w \frac{\partial v}{\partial x_{i}} dΩ [/itex]

    I have two examples in polar coordinates
    In first example I have [itex]\vec{n}=\vec{n_{r}}[/itex]

    [itex] \int_{\Gamma} \frac{1}{r^{2}} \frac{\partial^{2}w}{\partial\varphi^{2}}\frac{∂ v}{\partial r} \vec{n_{r}} d\Gamma = -2 \int_{Ω}\frac{1}{r^{3}}\frac{\partial^{2} w}{\partial \varphi^{2}} \frac{∂v}{∂r} dΩ + \int_{Ω}\frac{1}{r^{2}}\frac{\partial^{3} w}{∂r \partial \varphi^{2} } \frac{∂v}{∂r} dΩ + \int_{Ω}\frac{1}{r^{2}}\frac{\partial^{2} w}{\partial \varphi^{2} } \frac{∂^{2}v}{∂r^{2}} dΩ [/itex]

    and in second [itex]\vec{n}=\vec{n_{\varphi}}[/itex]

    [itex] \int_{\Gamma} \frac{1}{r^{2}} \frac{\partial w}{\partial\varphi}\frac{∂ v}{\partial r} \vec{n_{\varphi}} d\Gamma = \int_{Ω}\frac{1}{r^{3}}\frac{\partial^{2} w}{\partial \varphi^{2}} \frac{∂v}{∂r} dΩ + \int_{Ω}\frac{1}{r^{3}}\frac{\partial w}{\partial \varphi} \frac{∂^{2}v}{∂r∂\varphi} dΩ [/itex]

    When I integrate with respect to [itex]\varphi[/itex] I multiply equation by [itex]\frac{1}{r}[/itex] but I am no sure if this is correct.

    Are this two solutions correct?

    Thanks for answers
  2. jcsd
  3. Jan 24, 2013 #2
    Functions w and v are functions of r and [itex]\varphi[/itex] ( w = w(r, [itex]\varphi[/itex]) and v = v(r, [itex]\varphi[/itex]))
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